Friday, May 27, 2005

Looking Back (EDITED)

I was playing hopscotch with mathematicial blogs today and I stumbled upon an article written by a professor at UC Berkeley. It concerns undergraduate education for those math majors who do not proceed to graduate school.

(The article can be found here in PDF format.)

Mathematics without graduate school?
Heresy, you say?

I admit, that was my first inclination and now I fear I've been fully indoctrinated by academia. But that's fine, because I'm not arguing about that today. In fact, I'm not arguing anything at all: just reminiscing.

my education. Most of you already know the story of how I fell into the field of mathematics, so I won't retell it. But as for those years of enrolling in classes, I've realized something.

My early study of mathematics was hardly circumspect. As an undergrad I took exactly one abstract algebra course. I never took topology or number theory or logic / set theory, and if you asked me what Galois theory or representation theory or cohomology was, you'd receive a blank look in return.

The more I think about it, the more I realize that I learned very specific topics for a very specific purpose. I was still dabbling in the prerequisites in my sophomore year of college, stuff like Advanced Calculus I & II, Linear Algebra, ODE, and Abstract Algebra.

It was later that summer that my Linear Algebra prof offered me a research project and threw me a problem. Read this paper and learn what you need to understand it, he suggested. Ask me questions, and read. When you're ready, we'll test the technique on another case. Maybe we'll prove something new.

Forget the vector bundle approach (as if I knew then what it was). I was introduced to Riemannian Geometry from the concrete point of view of matrix theory and inner products, then the far newer subject of sub-Riemannian geometry from a control-theoretic viewpoint. I never realized geodesics and extremal problems would be that interesting.

The technique never worked, and the problem is still open. But we wouldn't know that until three years later.

My prof left on sabbatical but suggested that we can think about it a year from now, but there was much more to know before we could tackle other methods. When I asked him what sort of topics, he told me and I chose my junior and senior year curricula.

Complex Analysis. Real Analysis I & II. Functional Analysis. PDE I and Topics in PDE. The Calculus of Variations. Differential Geometry. Linear Operators. I read some topology on my own, and tried to understand why Sobolev spaces were the right kind of object to study.

As a result I've always thought of myself as an analyst-in-training and a problem solver. Theory and theory-building don't come natural to me, and every so often I catch myself thinking in terms of computations and technical matters. Now in a different school and a different place, the rules are different and I swear I've learned the mathematics that I missed and relearned all the mathematics that I thought I knew.

I don't think that sort of education was a waste. Every time I studied something new I knew what motivated me to do so .. but later the reasons changed, and I became an academic. Learning for learning's sake.

Why Graduate School? I can't remember if there was a distinct moment when I realized that I wanted to go to graduate school in this field. I wonder if there was ever such a moment. I'm tempted to say that "it just happened."

That's all well and good, but for those of us who've gone through the onslaught of forms and applications, deciding to go to graduate school requires a fair bit of thought and effort. There is the general GRE and the mathematics GRE, the jerryrigging of presenting yourself on paper, and the search for faculty who are willing to aggrandize you to other faculty. We ponder what qualifying exams and oral preliminary exams are like, what research is like, what the job market is like ..

.. but before all that, what five more years of schooling are like.

I guess I was just motivated early on. Someone showed me a seemingly simple problem, how complicated it becomes, and how quickly it does so. Perhaps it is my pride in humanity and its capacity for progress which has led me here: some problems are meant to be solved and someone must do so. If it means that I must become a mathematician, then I shall and so be it.

I wouldn't have it any other way.


Anonymous said...

Would you agree with the following statement: "people who won't go to grad school have no need to study math"?

janus said...

I'm going to interpret your statement as: people who won't go to graduate school in mathematics have no need to study math. If this is not the spirit of your comment then I apologize, but I cannot vouch for fields other than my own.

Next, I don't think I'm qualified to say how much the average college student should study math. It varies by discipline and by choice of career (though that isn't easily determined in advance), but I would like to think that the average student should acquire some appreciation and flavor of what mathematics is other than an introductory sequence in single-variable calculus.

That leaves the mathematics majors who know they will not proceed to graduate school. By their very choice they have to study some mathematics, and I believe the question is: what type of mathematics and how much?

This is a hard question to answer, but it isn't too hard in some cases. For those who wish to become actuaries or educators in K-12 schools, then it might be best to have a more specific curriculum to suit their needs.

Thinking more about it, it would be nice if mathematics departments offer information (possibly workshops) on what sort of careers are available for college graduates with a background in mathematics. It is important to know what your goal is and plan accordingly for it.

The general mathematics major should study a variety of topics, and this does depend on pure or applied maths. My essential idea is that they should be aware of what the field is, have a general understanding about sub-fields of study (basic facts and theorems) and the techniques developed for such study.

For pure majors, I would suggest an introduction of equal treatment to some core areas, such as geometry, analysis, algebra, and topology. They should also have some knowledge of more specific areas such as number theory, ODE, and combinatorics.

From there, if certain areas catch their interest then they should be free to pursue them. Perhaps a requirement of a fixed number of upper-level courses will do, but there is no need to demand variety: that is what the introduction is for.

I should not say anything about applied mathematics because I don't know how that field works. However, let me reiterate a previous point: I believe it is important that applied mathematics majors have a sense of what careers in industry are like -- possibly an internship program or co-op (as the engineers do).

In this way they can determine how their coursework should be fitted to their future, and this is not isolated to mathematics: perhaps they'll find that computer programming or some background in materials science will also be helpful, and they should be allowed this flexibility.

I could be wrong about many of these ideas; I'm just a graduate student, not an undergraduate chair for a mathematics department. If any of what I've said is unreasonable, then please say so.

fragments of angry candy said...

but good to think ahead, Jasun--maybe someday you will be an undergraduate chair for a math department.

i had wu for complex analysis as an undergrad. now it makes sense that ideas like those in the paper were brewing in his head at the time (I guess then he was also teaching his experimental courses. I'm not sure mine was an experimental course because it was numbered as the standard requirement, but it wasn't honors or anything, maybe that was the grad school track.) he had a tendency to lecture on broader things, and wanted us all to buy this book, "what is mathematics." i bought it, but confess i don't think i ever got around to reading it. he also had a reputation for grading hard and gave me the worst grade for all my undergraduate time.

it may be obvious to some that as an undergrad i didn't know that i would go to grad school in math. but i wouldn't have liked to exclude any possibilities. if two versions of a course were offered i would have probably sat in on both and went with my gut on which one to pick. i guess i would have been likely to pick the one aimed at grad school wannabes since that's where i found myself eventually, but then again, intent is one thing, result another when it comes to teaching, so my choice would still heavily depend on that intangible gut feel i get from a class during the first week.